Statistics & Probability Letters 54 (2001) 75–78 Path continuity of the nonlinear lter Abhay G. Bhatt ∗ , Rajeeva L. Karandikar Indian Statistical Institute, 7, S.J.S. Sansanwal Marg, New Delhi 110016, India Received July 2000; received in revised form March 2001 Abstract We consider the nonlinear ltering model with signal and observation noise independent, and show that in case the signal is continuous in probability, the lter admits a version whose paths are continuous. The analysis is based on expressing the nonlinear lter as a Wiener functional via the Kallianpur–Striebel Bayes formula. c  2001 Elsevier Science B.V. All rights reserved MSC: primary 60G35; 62M20; secondary 60G17; 60G44 Keywords: Nonlinear ltering; Path continuity 1. Introduction Consider the nonlinear ltering model Y t =  t 0 h(X s )ds + W t ; 06t 6T; (1.1) where X is the signal process, assumed to take values in a complete separable metric space E and having r.c.l.l. paths, the observation noise W is assumed to be an R k valued Brownian motion independent of the signal X , h is a measurable function and Y is the observation process. The optimal lter  t is given by 〈 t ;f〉 = E[f(X t )|F Y t ]; ∀f ∈ C b (E): (1.2) Here C b (E) is the class of bounded continuous functions on E, the processes X and W are dened on a probability space (; F ;P) and F Y t = {Y s :06s6t } is the observation -eld. The function h is assumed to satisfy  T 0 |h(X s )| 2 ds¡∞ a:s: [P]: (1.3) * Corresponding author. Tel.: +91-11-651-6200; fax: +91-11-685-6779. E-mail address: abhay@isid.ac.in (A.G. Bhatt). 0167-7152/01/$-see front matter c  2001 Elsevier Science B.V. All rights reserved PII:S0167-7152(01)00068-2